There are differences in meaning.
I didn’t get enough sleep. (Not long enough for me to function my best)
I didn’t sleep well. (Maybe the length of time, maybe a poor quality. Judged on an absolute scale)
I didn’t get my full night’s sleep. (Last night, I expected to get a certain quantity, but didn’t. It could still be enough. It could still be a pretty good night’s sleep. It might not be)
‘I haven’t slept very much’ could be natural but is a little more ambiguous as to the time period. I’d be more likely to use it if the period of time is longer.
Since my husband lost his job, I haven’t slept very much. I lie awake at night worrying about our finances.